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Earl W. Swokowski
Chapter 6, Problem 26
Set up sums of integrals that can be used to find the area of the shaded region by integrating with respect to $|a| x$ and $|b| y$
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Official textbook answer
Video by David Marsella
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Express the area of the following shaded regions in terms of (a) one or more integrals with respect to $x,$ and
(b) one or more integrals with respec…
So here we have. Why equals X squared minus two and y equals X, And we want to find the area of the shaded region. But we’ve been asked to just set it up. We’re just going to set up the integral we’re gonna set it up for both. With respect to X and with respect alive, we don’t actually have to solve the integral, which is gonna save us a lot of time, actually. So first, let’s look at this. We know that our boundaries are the X axis, the line y equals X in our parabola X squared minus two. And we know we’re gonna need these three points to help set our boundaries for any girls. So let’s go about finding those points first. We know this point is 00 because y equals X passes through the origin. That was easy enough. We’re gonna need this interception point where y equals X squared minus two passes the x axis. So if we set y equals zero there we get zero equals X squared minus two, which factors? Two X plus radical too and X minus radical, too. So we have the excess intercept at X equals plus or minus groups, plus or minus radical, too. Now, our point is in the negatives we’re not We’re not worried about this point here in the…